Theorem laut11

Description: One-to-one property of a lattice automorphism. (Contributed by NM, 20-May-2012)

	Ref	Expression
Hypotheses	laut1o.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	laut1o.i	⊢ 𝐼 = ( LAut ‘ 𝐾 )
Assertion	laut11	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑋 ) = ( 𝐹 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		laut1o.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		laut1o.i	⊢ 𝐼 = ( LAut ‘ 𝐾 )
3	1 2	laut1o	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 )
4		f1of1	⊢ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 → 𝐹 : 𝐵 –1-1→ 𝐵 )
5	3 4	syl	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ) → 𝐹 : 𝐵 –1-1→ 𝐵 )
6		f1fveq	⊢ ( ( 𝐹 : 𝐵 –1-1→ 𝐵 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑋 ) = ( 𝐹 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) )
7	5 6	sylan	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑋 ) = ( 𝐹 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) )